"Uncle Penn" - Original by Bill Monroe, covered by Ricky Skaggs
"Third Rate Romance, Low Rent Rendezvous" - Original by The Amazing Rhythm Aces, covered by Sammy Kershaw
"Whiskey in the Jar" - Original by Thin Lizzy, ass kicked by Metallica
Don't know if this counts, but
"Walk This Way" - Original by Aerosmith, covered by Run DMC/Aerosmith
Worst cover by someone I usually like,
"Uncle John's Band" - Grateful Dead, covered by Jimmy Buffet